Gauss Hermite

Example

use integrate::gauss_quadrature::gauss_hermite_rule;

let f = |x: f64| 1.0;

let n:usize = 100;

let integral = gauss_hermite_rule(f, n);
println!("{}",integral);

Understanding Gauss-Hermite rule

Gauss-Hermite quadrature formulas are used to integrate functions \(f(x) e^{x^2}\) from \(-\infty\) to \(+\infty\).

With respect to the inner product

\[ \langle f,g \rangle = \int_{-\infty}^{+\infty} f(x) * g(x) * w(x) dx \]

the Hermite polynomials

\[ H_n(x) = (-1)^n * e^{x^2}* \frac{\partial^{n} e^{-x^2}}{\partial x^n} \quad \text{for} \quad n > 0 \]

and \(H_0(x) = 1\) form an orthogonal family of polynomials with weight function \(w(x) = e^{-x^2}\) on the entire \(x\)-axis.

The \(n\)-point Gauss-Hermite quadrature formula, \(GH_n ( f(x) )\), for approximating the integral of \(f(x) e^{-x^2}\) over the entire \(x\)-axis, is given by

\[ GH_n ( f(x) ) = A_1 f(x_1) + ··· + A_n f(x_n) \]

where \(x_i\) , for \(i = 1,...,n\), are the zeros of \(H_n\) and

\[ A_i = \frac{2^{n+1} * n! * \sqrt{\pi}}{H_{n-1} (x_i)^2} \quad \text{for} \quad i = 1,...,n \]